Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,2,Mod(21,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 81.5
Root \(0.455023i\) of defining polynomial
Character \(\chi\) \(=\) 85.81
Dual form 85.2.e.a.21.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.783476i q^{2} +(0.385357 + 0.385357i) q^{3} +1.38617 q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.301918 + 0.301918i) q^{6} +(-0.840380 + 0.840380i) q^{7} +2.65298i q^{8} -2.70300i q^{9} +(0.554001 - 0.554001i) q^{10} +(-1.80608 + 1.80608i) q^{11} +(0.534168 + 0.534168i) q^{12} -0.368117 q^{13} +(-0.658417 - 0.658417i) q^{14} -0.544977i q^{15} +0.693788 q^{16} +(-2.46531 - 3.30488i) q^{17} +2.11773 q^{18} -6.61138i q^{19} +(-0.980167 - 0.980167i) q^{20} -0.647692 q^{21} +(-1.41502 - 1.41502i) q^{22} +(-2.73186 + 2.73186i) q^{23} +(-1.02234 + 1.02234i) q^{24} +1.00000i q^{25} -0.288411i q^{26} +(2.19769 - 2.19769i) q^{27} +(-1.16491 + 1.16491i) q^{28} +(-1.63466 - 1.63466i) q^{29} +0.426976 q^{30} +(4.68480 + 4.68480i) q^{31} +5.84952i q^{32} -1.39197 q^{33} +(2.58929 - 1.93151i) q^{34} +1.18848 q^{35} -3.74681i q^{36} +(2.24619 + 2.24619i) q^{37} +5.17986 q^{38} +(-0.141856 - 0.141856i) q^{39} +(1.87594 - 1.87594i) q^{40} +(-5.16572 + 5.16572i) q^{41} -0.507451i q^{42} +6.82350i q^{43} +(-2.50353 + 2.50353i) q^{44} +(-1.91131 + 1.91131i) q^{45} +(-2.14034 - 2.14034i) q^{46} +7.80793 q^{47} +(0.267356 + 0.267356i) q^{48} +5.58752i q^{49} -0.783476 q^{50} +(0.323534 - 2.22358i) q^{51} -0.510272 q^{52} -8.01219i q^{53} +(1.72184 + 1.72184i) q^{54} +2.55419 q^{55} +(-2.22951 - 2.22951i) q^{56} +(2.54774 - 2.54774i) q^{57} +(1.28072 - 1.28072i) q^{58} +5.22381i q^{59} -0.755428i q^{60} +(5.74267 - 5.74267i) q^{61} +(-3.67043 + 3.67043i) q^{62} +(2.27155 + 2.27155i) q^{63} -3.19538 q^{64} +(0.260298 + 0.260298i) q^{65} -1.09058i q^{66} -7.94564 q^{67} +(-3.41733 - 4.58111i) q^{68} -2.10548 q^{69} +0.931143i q^{70} +(8.40610 + 8.40610i) q^{71} +7.17100 q^{72} +(-10.4176 - 10.4176i) q^{73} +(-1.75984 + 1.75984i) q^{74} +(-0.385357 + 0.385357i) q^{75} -9.16447i q^{76} -3.03559i q^{77} +(0.111141 - 0.111141i) q^{78} +(0.575011 - 0.575011i) q^{79} +(-0.490582 - 0.490582i) q^{80} -6.41521 q^{81} +(-4.04721 - 4.04721i) q^{82} -3.99116i q^{83} -0.897809 q^{84} +(-0.593666 + 4.08014i) q^{85} -5.34604 q^{86} -1.25986i q^{87} +(-4.79150 - 4.79150i) q^{88} +9.14311 q^{89} +(-1.49746 - 1.49746i) q^{90} +(0.309359 - 0.309359i) q^{91} +(-3.78681 + 3.78681i) q^{92} +3.61064i q^{93} +6.11732i q^{94} +(-4.67495 + 4.67495i) q^{95} +(-2.25415 + 2.25415i) q^{96} +(-4.99529 - 4.99529i) q^{97} -4.37769 q^{98} +(4.88185 + 4.88185i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 12 q^{4} - 4 q^{10} - 4 q^{11} - 8 q^{12} - 4 q^{14} + 4 q^{16} + 12 q^{17} + 28 q^{18} - 8 q^{20} - 16 q^{21} + 20 q^{22} + 12 q^{23} + 4 q^{24} - 4 q^{27} + 4 q^{28} - 12 q^{29} - 8 q^{30}+ \cdots + 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.783476i 0.554001i 0.960870 + 0.277000i \(0.0893404\pi\)
−0.960870 + 0.277000i \(0.910660\pi\)
\(3\) 0.385357 + 0.385357i 0.222486 + 0.222486i 0.809544 − 0.587059i \(-0.199714\pi\)
−0.587059 + 0.809544i \(0.699714\pi\)
\(4\) 1.38617 0.693083
\(5\) −0.707107 − 0.707107i −0.316228 − 0.316228i
\(6\) −0.301918 + 0.301918i −0.123257 + 0.123257i
\(7\) −0.840380 + 0.840380i −0.317634 + 0.317634i −0.847858 − 0.530224i \(-0.822108\pi\)
0.530224 + 0.847858i \(0.322108\pi\)
\(8\) 2.65298i 0.937970i
\(9\) − 2.70300i − 0.901000i
\(10\) 0.554001 − 0.554001i 0.175190 − 0.175190i
\(11\) −1.80608 + 1.80608i −0.544555 + 0.544555i −0.924861 − 0.380306i \(-0.875819\pi\)
0.380306 + 0.924861i \(0.375819\pi\)
\(12\) 0.534168 + 0.534168i 0.154201 + 0.154201i
\(13\) −0.368117 −0.102097 −0.0510487 − 0.998696i \(-0.516256\pi\)
−0.0510487 + 0.998696i \(0.516256\pi\)
\(14\) −0.658417 − 0.658417i −0.175969 − 0.175969i
\(15\) − 0.544977i − 0.140712i
\(16\) 0.693788 0.173447
\(17\) −2.46531 − 3.30488i −0.597926 − 0.801552i
\(18\) 2.11773 0.499155
\(19\) − 6.61138i − 1.51676i −0.651816 − 0.758378i \(-0.725992\pi\)
0.651816 − 0.758378i \(-0.274008\pi\)
\(20\) −0.980167 − 0.980167i −0.219172 − 0.219172i
\(21\) −0.647692 −0.141338
\(22\) −1.41502 − 1.41502i −0.301684 − 0.301684i
\(23\) −2.73186 + 2.73186i −0.569632 + 0.569632i −0.932025 − 0.362394i \(-0.881960\pi\)
0.362394 + 0.932025i \(0.381960\pi\)
\(24\) −1.02234 + 1.02234i −0.208685 + 0.208685i
\(25\) 1.00000i 0.200000i
\(26\) − 0.288411i − 0.0565621i
\(27\) 2.19769 − 2.19769i 0.422945 − 0.422945i
\(28\) −1.16491 + 1.16491i −0.220147 + 0.220147i
\(29\) −1.63466 − 1.63466i −0.303549 − 0.303549i 0.538851 − 0.842401i \(-0.318858\pi\)
−0.842401 + 0.538851i \(0.818858\pi\)
\(30\) 0.426976 0.0779548
\(31\) 4.68480 + 4.68480i 0.841415 + 0.841415i 0.989043 − 0.147628i \(-0.0471639\pi\)
−0.147628 + 0.989043i \(0.547164\pi\)
\(32\) 5.84952i 1.03406i
\(33\) −1.39197 −0.242311
\(34\) 2.58929 − 1.93151i 0.444060 − 0.331251i
\(35\) 1.18848 0.200889
\(36\) − 3.74681i − 0.624468i
\(37\) 2.24619 + 2.24619i 0.369271 + 0.369271i 0.867211 − 0.497940i \(-0.165910\pi\)
−0.497940 + 0.867211i \(0.665910\pi\)
\(38\) 5.17986 0.840284
\(39\) −0.141856 − 0.141856i −0.0227152 − 0.0227152i
\(40\) 1.87594 − 1.87594i 0.296612 − 0.296612i
\(41\) −5.16572 + 5.16572i −0.806749 + 0.806749i −0.984140 − 0.177391i \(-0.943234\pi\)
0.177391 + 0.984140i \(0.443234\pi\)
\(42\) − 0.507451i − 0.0783014i
\(43\) 6.82350i 1.04057i 0.853992 + 0.520287i \(0.174175\pi\)
−0.853992 + 0.520287i \(0.825825\pi\)
\(44\) −2.50353 + 2.50353i −0.377422 + 0.377422i
\(45\) −1.91131 + 1.91131i −0.284921 + 0.284921i
\(46\) −2.14034 − 2.14034i −0.315576 − 0.315576i
\(47\) 7.80793 1.13890 0.569452 − 0.822025i \(-0.307156\pi\)
0.569452 + 0.822025i \(0.307156\pi\)
\(48\) 0.267356 + 0.267356i 0.0385895 + 0.0385895i
\(49\) 5.58752i 0.798218i
\(50\) −0.783476 −0.110800
\(51\) 0.323534 − 2.22358i 0.0453038 − 0.311364i
\(52\) −0.510272 −0.0707620
\(53\) − 8.01219i − 1.10056i −0.834980 − 0.550280i \(-0.814521\pi\)
0.834980 − 0.550280i \(-0.185479\pi\)
\(54\) 1.72184 + 1.72184i 0.234312 + 0.234312i
\(55\) 2.55419 0.344407
\(56\) −2.22951 − 2.22951i −0.297931 − 0.297931i
\(57\) 2.54774 − 2.54774i 0.337456 − 0.337456i
\(58\) 1.28072 − 1.28072i 0.168167 − 0.168167i
\(59\) 5.22381i 0.680082i 0.940411 + 0.340041i \(0.110441\pi\)
−0.940411 + 0.340041i \(0.889559\pi\)
\(60\) − 0.755428i − 0.0975253i
\(61\) 5.74267 − 5.74267i 0.735273 − 0.735273i −0.236386 − 0.971659i \(-0.575963\pi\)
0.971659 + 0.236386i \(0.0759630\pi\)
\(62\) −3.67043 + 3.67043i −0.466145 + 0.466145i
\(63\) 2.27155 + 2.27155i 0.286188 + 0.286188i
\(64\) −3.19538 −0.399423
\(65\) 0.260298 + 0.260298i 0.0322860 + 0.0322860i
\(66\) − 1.09058i − 0.134241i
\(67\) −7.94564 −0.970715 −0.485357 − 0.874316i \(-0.661310\pi\)
−0.485357 + 0.874316i \(0.661310\pi\)
\(68\) −3.41733 − 4.58111i −0.414412 − 0.555542i
\(69\) −2.10548 −0.253470
\(70\) 0.931143i 0.111293i
\(71\) 8.40610 + 8.40610i 0.997621 + 0.997621i 0.999997 − 0.00237624i \(-0.000756380\pi\)
−0.00237624 + 0.999997i \(0.500756\pi\)
\(72\) 7.17100 0.845111
\(73\) −10.4176 − 10.4176i −1.21929 − 1.21929i −0.967881 − 0.251409i \(-0.919106\pi\)
−0.251409 − 0.967881i \(-0.580894\pi\)
\(74\) −1.75984 + 1.75984i −0.204577 + 0.204577i
\(75\) −0.385357 + 0.385357i −0.0444972 + 0.0444972i
\(76\) − 9.16447i − 1.05124i
\(77\) − 3.03559i − 0.345938i
\(78\) 0.111141 − 0.111141i 0.0125843 − 0.0125843i
\(79\) 0.575011 − 0.575011i 0.0646938 − 0.0646938i −0.674020 − 0.738713i \(-0.735434\pi\)
0.738713 + 0.674020i \(0.235434\pi\)
\(80\) −0.490582 − 0.490582i −0.0548488 − 0.0548488i
\(81\) −6.41521 −0.712801
\(82\) −4.04721 − 4.04721i −0.446940 − 0.446940i
\(83\) − 3.99116i − 0.438087i −0.975715 − 0.219044i \(-0.929706\pi\)
0.975715 − 0.219044i \(-0.0702937\pi\)
\(84\) −0.897809 −0.0979590
\(85\) −0.593666 + 4.08014i −0.0643921 + 0.442554i
\(86\) −5.34604 −0.576479
\(87\) − 1.25986i − 0.135071i
\(88\) −4.79150 − 4.79150i −0.510776 − 0.510776i
\(89\) 9.14311 0.969168 0.484584 − 0.874745i \(-0.338971\pi\)
0.484584 + 0.874745i \(0.338971\pi\)
\(90\) −1.49746 − 1.49746i −0.157847 − 0.157847i
\(91\) 0.309359 − 0.309359i 0.0324296 − 0.0324296i
\(92\) −3.78681 + 3.78681i −0.394802 + 0.394802i
\(93\) 3.61064i 0.374406i
\(94\) 6.11732i 0.630953i
\(95\) −4.67495 + 4.67495i −0.479640 + 0.479640i
\(96\) −2.25415 + 2.25415i −0.230063 + 0.230063i
\(97\) −4.99529 − 4.99529i −0.507195 − 0.507195i 0.406470 − 0.913664i \(-0.366760\pi\)
−0.913664 + 0.406470i \(0.866760\pi\)
\(98\) −4.37769 −0.442213
\(99\) 4.88185 + 4.88185i 0.490644 + 0.490644i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 85.2.e.a.81.5 yes 12
3.2 odd 2 765.2.k.b.676.2 12
4.3 odd 2 1360.2.bt.d.81.2 12
5.2 odd 4 425.2.j.c.149.2 12
5.3 odd 4 425.2.j.b.149.5 12
5.4 even 2 425.2.e.f.251.2 12
17.2 even 8 1445.2.a.o.1.2 6
17.4 even 4 inner 85.2.e.a.21.2 ✓ 12
17.8 even 8 1445.2.d.g.866.9 12
17.9 even 8 1445.2.d.g.866.10 12
17.15 even 8 1445.2.a.n.1.2 6
51.38 odd 4 765.2.k.b.361.5 12
68.55 odd 4 1360.2.bt.d.1041.2 12
85.4 even 4 425.2.e.f.276.5 12
85.19 even 8 7225.2.a.z.1.5 6
85.38 odd 4 425.2.j.c.174.2 12
85.49 even 8 7225.2.a.bb.1.5 6
85.72 odd 4 425.2.j.b.174.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.e.a.21.2 ✓ 12 17.4 even 4 inner
85.2.e.a.81.5 yes 12 1.1 even 1 trivial
425.2.e.f.251.2 12 5.4 even 2
425.2.e.f.276.5 12 85.4 even 4
425.2.j.b.149.5 12 5.3 odd 4
425.2.j.b.174.5 12 85.72 odd 4
425.2.j.c.149.2 12 5.2 odd 4
425.2.j.c.174.2 12 85.38 odd 4
765.2.k.b.361.5 12 51.38 odd 4
765.2.k.b.676.2 12 3.2 odd 2
1360.2.bt.d.81.2 12 4.3 odd 2
1360.2.bt.d.1041.2 12 68.55 odd 4
1445.2.a.n.1.2 6 17.15 even 8
1445.2.a.o.1.2 6 17.2 even 8
1445.2.d.g.866.9 12 17.8 even 8
1445.2.d.g.866.10 12 17.9 even 8
7225.2.a.z.1.5 6 85.19 even 8
7225.2.a.bb.1.5 6 85.49 even 8